229-233] Spherical Harmonics 203 



of the radius and origin of inversion, we can give any radii we like to the 

 two spheres. 



If we take the radius of one to be infinite, we get the distribution on a 

 plane with an excrescence in the form of a piece of a sphere : in the particu- 

 lar case of n = 2, this excrescence is hemispherical, and we obtain the 

 distribution of electricity on a plane face with a hemispherical boss. This 

 can, however, be obtained more directly by the method of 219. 



SPHERICAL HARMONICS. 



232. The problem of finding the solution of any electrostatic problem is 

 equivalent to that of finding a solution of Laplace's equation 



V 2 F=0 



throughout the space not occupied by conductors, such as shall satisfy certain 

 conditions at the boundaries of this space i.e. at infinity and on the surfaces 

 of conductors. The theory of spherical harmonics attempts to provide a 

 general solution of the equation V 2 F = 0. 



This is no convenient general solution in finite terms : we therefore 

 examine solutions expressed as an infinite series. If each term of such 

 a series is a solution of the equation, the sum of the series is necessarily 

 a solution. 



233. Let us take spherical polar coordinates r, 0, 0, and search for 

 solutions of the form 



V=RS, 



where R is a function of r only, and 8 is a function of 6 and </> only. 



Laplace's equation, expressed in spherical polars, can be obtained analyti- 

 cally from the equation 



1 ^ + = 



by changing variables from x, y, z to r, 6, (f>, but is most easily obtained by 

 applying Gauss' theorem to the small element of volume bounded by the 

 spheres r and r -f dr, the cones 6 and 9 -H dO, and the diametral planes <f> and 

 <j) + dcf>. The equation is found to be 



* 



r 2 sin<9 W 

 Substituting the value =218, we obtain 



r* dr 

 or, simplifying, 



d 3\ R 9 / . dS\ R d^S _ 



) * " 



1 9 / dR\ 1 8 / . ,. dS\ 1 d 2 S 



I r - 1 -\ [ sin (/ i H = 0. 



R dr \ dr 1 $sin# W \ W) 8 sin 2 6 dd>' 2 



